Electronic Journal of Probability

Isotropic local laws for sample covariance and generalized Wigner matrices

Bloemendal Alex, László Erdős, Antti Knowles, Horng-Tzer Yau, and Jun Yin

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Abstract

We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M \times N$ matrix with independent random entries.  We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent $(X^* X - z)^{-1}$ converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity $\langle v , (X^* X - z)^{-1}w\rangle - \langle v , w\rangle m(z)$, where $m$ is the Stieltjes transform of the Marchenko-Pastur law and $v , w \in \mathbb{C}^N$. We require the logarithms of the dimensions $M$ and $N$ to be comparable. Our result holds down to scales $\Im z \geq N^{-1+\varepsilon}$ and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 33, 53 pp.

Dates
Accepted: 15 March 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065675

Digital Object Identifier
doi:10.1214/EJP.v19-3054

Mathematical Reviews number (MathSciNet)
MR3183577

Zentralblatt MATH identifier
1288.15044

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Alex, Bloemendal; Erdős, László; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun. Isotropic local laws for sample covariance and generalized Wigner matrices. Electron. J. Probab. 19 (2014), paper no. 33, 53 pp. doi:10.1214/EJP.v19-3054. https://projecteuclid.org/euclid.ejp/1465065675


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